This expository article is based on a homonymous talk I gave during the "II Encuentro de Geometría" which took place in La Falda, Sierras de Córdoba, from June 6th to 11th of 2005. It summarizes previous results from [MP], [MP2], [MPR], and [Po], answering standard questions in spectral geometry by using a special class of compact Riemannian manifolds.

Spectral Geometry. It is a kind of mixture between Spectral Theory and Riemannian Geometry. The general situation is to consider (pseudo) differential operators acting on sections of bundles of Riemannian manifolds. However, one usually considers a compact Riemannian manifold , a vector bundle and an elliptic self-adjoint differential operator acting on smooth sections of , i.e. . Since is compact, has a discrete spectrum, denoted by , consisting of real eigenvalues of finite multiplicity which accumulate only at infinity. In symbols, we have

, .

, -eigenspace.

We can also think of as being the set .

Two manifolds are called isospectral with respect to, or simply -isospectral, if . That is, if have the same set of eigenvalues with the same corresponding multiplicities. It is a general fact that the spectrum determines the dimension and the volume of . In other words, if are -isospectral, then and . The spectrum is said to be asymmetric if for some .

Main Aim. The goal of Spectral Geometry is to study the spectrum of and the interrelations between this object and the geometry or topology of . That is, knowing the spectrum, what can be said geometrically about ? Conversely, which spectral data can be deduced provided that we know the geometry of ? This can be summarize in the following diagram

Incidentally, by using the diagonal "maps" one could say something about the horizontal "map".

Main Problems. There are several different ways of studying the spectrum. In my opinion, the following are the three most important and interesting ones. In this paper we shall collect results concerning all of them. Computation of . The problem is to determine the eigenvalues and their multiplicities . This is in general a difficult task in the sense that this cannot always be done. Indeed, there are few classes of manifolds with explicitly known spectrum for some given operator. The simplest case is the Laplacian acting on smooth functions on the torus .

Isospectrality. Physically, it is a problem with more than a century old and inquires about the possibility of changing shape while sounding the same. Mathematically, it begun in 1964 with the famous Kac's question Can one hear the shape of a drum? ([Ka]) and the negative answer given by Milnor to a related question ([Mi]). There are basically two antagonistic approaches to this problem: criteria vs. counterexamples. In the first case, one seeks sufficient conditions ensuring that two manifolds are isospectral. This is what some people have called Optimistic Spectral Geometry. On the contrary, in the second case, one tries to produce examples of pairs of isospectral manifolds which are very similar to each other but differing in some geometrical or topological property . In this case, we say that this particular property cannot be heard or that we cannot hear property . This has been fairly called Pessimistic Spectral Geometry, but I would faintly call it DeafferentialGeometry. One interesting challenge here is to construct big families (the bigger the best) with respect to the dimension , of isospectral -manifolds which are topologically very different (the more different the best) to each other. One purpose of this might be to tightly highlight the fact that if some property cannot be heard, it is not merely an isolated casualty but a concrete reality we cannot ignore.

Spectral asymmetry: following the acoustic jargon before, one studies now when our drum (the manifold) is out of tune. That is, when the positive and the negative spectra differ. The usual devices designed to detect this phenomenon are the eta series and the eta invariant. One wants to compute them explicitly.

Summary of results. Here we give a list of the principal results obtained for -manifolds (that is, compact flat manifolds having holonomy group isomorphic to ) relative to the problems mentioned before. The results will be properly stated and explained in the body of the paper.

A. Full Laplacian : (1) all -manifolds covered by the same torus (or by isospectral tori) are isospectral on differential forms of mixed degree; (2) There are big families of -isospectral manifolds.

B. Spin structures: (3) we give necessary and sufficient conditions for their existence; (4) there are families of -manifolds which are spin while there are others which are not spin; (5) we answer Webb's question: "Can one hear the property of being spin on a compact Riemannianmanifold?".

C. Dirac spectrum: (6) we compute the multiplicities of the eigenvalues of twisted Dirac operators for an arbitrary spin -manifold.

D. Dirac isospectrality: (7) we obtain several examples of pairs of -isospectral manifolds having different topological, geometrical or spectral properties; (8) there are big families of -isospectral manifolds.

E. Spectral asymmetry. (9) we give a characterization of those manifolds having asymmetric Dirac spectrum; (10) explicit expressions for the eta series and the -invariant are given; (11) we answer Schueth's question: "Can one hear the-invariants of a compactRiemannian manifold?".

1. -manifolds

What are we talking about?. A Bieberbach group is a crystallographic group without torsion. That is, a discrete, cocompact, torsion-free subgroup of the isometries of . Such acts properly discontinuously on , thus is a compact flat Riemannian manifold having fundamental group . Any element decomposes uniquely as , where and denotes translation by .

By the classical Bieberbach's theorems we have the following two basic results: (i) the translations in form a normal maximal abelian subgroup of finite index, with a lattice in which is -stable for each (as usual, one identifies with ) and (ii) the restriction to of the canonical projection given by is a group homomorphism with kernel and is a finite subgroup of . It turns out that satisfies the exact sequence of groups

The group is called the holonomy group of and it is isomorphic to the linear holonomy group of the Riemannian manifold . The action of on by conjugation defines an integral representation which is usually called the integral holonomyrepresentation of . Note that this representation does not determine the group , i.e. there may be many non-isomorphic Bieberbach groups with the same holonomy representation.

A compact Riemannian manifold with holonomy group isomorphic to will be called an -manifold (see [Ch]). We shall only be concerned with -manifolds which, by the Cartan-Ambrose-Singer theorem, are necessarily flat.

Compact flat manifolds. A flat manifold is a closed, connected, Riemannian manifold , whose curvature identically vanishes. Notably, by the Killing-Hopf theorem any compact flat manifold is isometric to a quotient , with a Bieberbach group. Putting the Bieberbach theorem's into Riemannian language (see [Wo] or [Ch]) we get that: (i) is covered by the associated flat torus and the covering is a local isometry, (ii) is affinely equivalent to if and only if and (iii) there is a finite number of classes of affine equivalence of compact flat manifolds, in each dimension.

Up to equivalence, in dimension 1 there is only one compact flat manifold, the circle, while in dimension 2 there are two, the torus and the Klein bottle:

The number of compact flat manifolds grows rapidly with the dimension and a classification is unfortunately known only up to dimension 6.

There are two nice results concerning compact flat manifolds. One says that there are a plethora of them while the other says that all these manifolds bound: (1) Every finite groupcan be realized as the holonomy group of a compact flat manifold ([AK]) and (2) Ifis a compact flat-manifold then there is a compact-manifoldsuch that ([HR]).

Into the jungle. A -manifold is just a compact flat -manifold whose holonomy group is isomorphic to , with . Thus, it is of the form where , with and satisfying , , and , for .

Some friendly tribes. We now introduce some particularly interesting classes of -manifolds that will be used in the rest of the paper.

-manifolds. They generalize the Klein Bottle in the sense that they are quotients of tori divided by a -action. They are determined by the integral holonomy representation which can be parametrized by the block matrices

with and . The corresponding diffeomorphism classes are represented by , the canonical lattice. One can compute their first integral homology groups and their Betti numbers. Indeed, and , for . (See [MP]).

Primitive -manifolds. We recall that primitive means that , that is has trivial center. By a construction due to Calabi (see [Ca], [Wo]), any compact flat manifold can be obtained from a primitive one. Primitive -manifolds are also determined by the integral holonomy representation, which decomposes as a sum of integral representations of rank ([Ti]).

Diagonal type. A compact flat manifold is of diagonal type if there is an orthonormal -basis of satisfying , , for every . In this case we say that have diagonal holonomy representation. One can assume that is the canonical lattice and that . They necessarily have holonomy group . (See [MR3]).

Hantzsche-Wendt manifolds. (Or, HW-manifolds, for short). They are the orientable -manifolds in odd dimension . Any such manifold is given by where fixes , for , if , and (note that ). They generalize the only orientable -manifold existing in dimension 3, historically called the Hantzsche-Wendt manifold, and were studied in [MR]. They are primitive, of diagonal type and, furthermore, they are rational homology spheres, i.e. for every HW-manifold . Also, one can associate certain directed graphs to them.

Generalized Hantzsche-Wendt manifolds. (Or GHW-manifolds). They are simply the -manifolds in dimension . They share many properties with HW-manifolds but they are not primitive in general. There are different integral holonomy representations, all of diagonal type. (See [RS]).

A little bit of numerology. As we have seen, we have the following natural inclusions . In the table below we compare the cardinality of these families. We see that, at least in low dimensions, the class of -manifolds represents more than half of the compact flat manifolds.

2. The full Laplacian

Consider the Laplacian on -forms . It is a first order elliptic differential operator acting on smooth sections of the -exterior bundle of . The spectrum of this operator on compact flat manifolds was studied in [MR2] (see also [MR3], [MR4]). The multiplicity of the eigenvalue of has the expression

where , with , and is the character of the -exterior representation. For of diagonal type, this character is given by integer values of certain polynomials. In fact, for , we have

where and is the (binary) Krawtchouk polynomial of order and degree . They are discrete orthogonal polynomials (see [KL]). The first ones have the expressions , , , , etc.

From now on in this section we refer to [MPR]. One can simply define a Laplacian on arbitrary forms by considering the -Laplacians altogether, that is, we can take

This full Laplacian is again a first order elliptic differential operator which acts on sections of the full exterior bundle of . The eigenvalues are still of the form , but their multiplicities are now given by the sum . Clearly, -isospectrality (i.e. isospectrality with respect to ) for all implies -isospectrality, but the converse is far from being true, as will be shown in Example 2.2 below.

Thus, two-manifoldsare-isospectral if and only if the translationlatticesare isospectral. In particular, for fixedand, all -manifoldswith covering torusare-isospectral.

Sketch of proof. Let . Then, the 's diagonalize simultaneously with eigenvalues . Thus, every is conjugate in to the diagonal matrix where is the identity matrix in . Thus . Hence, we have and, adding over

Now, for , one can show that . Since if and only if , we finally get that , as asserted. □

Note 1. The proof seems to be of an entirely combinatorial nature since it only depends on (), and the Krawtchouk polynomials at integer values have some combinatorial interpretations in the literature.

Example 2.2 (-manifolds of dim 3). We illustrate the theorem in the simplest non-trivial case, i.e. when and . Up to diffeomorphism, there are only three -manifolds in dimension 3 (see [Wo]). They are and , in the notation of page 138.

In the tables below we give the multiplicities for , with , and also for , of the 2 lowest non trivial eigenvalues.

The values in the tables show that the manifolds are not -isospectral to each other for any . However, we can see how all these multiplicities balance, that is how they manage to distribute themselves in order to have equal sums for each eigenvalue, in each case.

We really need the hypothesisin the theorem. The "magical" averaging phenomenon, present when considering all the -Laplacians simultaneously, only seems to work in the case considered. The result does not hold, in general, for holonomy groups different from . There is a pair of 6-dimensional orientable -manifolds, , which are not -isospectral ([MPR, Ex. 3.5]), even though they are isospectral on functions. In fact, let and take and where , , , with Following [MR2] one can prove that are -isospectral (and hence -isospectral, by orientability) but they are not -isospectral for . Since we have that . By [Hi], the Betti numbers , , for and are respectively given by 1,2,5,8,5,2,1 and 1,2,3,4,3,2,1. In this way we get that while , hence are not isospectral on forms.

Note 2. One can also consider the Laplacian on even/odd forms given by and . The results in Theorem 2.1 hold mutatis mutandis but now with . In particular, .

Big families of-isospectral manifolds. As a straight consequence of the result in the previous theorem, we can exhibit several big families of -isospectral -manifolds in arbitrary dimension . Here, big alludes to the fact that the cardinality of the family grows polynomially —or even exponentially— with respect to and, also, that all the manifolds in each family are not homeomorphic to each other. It is worth noting that for the Laplacian on functions, or on -forms, there are not known examples of such exponential families. In the famous isospectral deformations of Gordon and Wilson (see [GW]), the manifolds have different metrics but they are all homeomorphic to each other.

Consider the following families: , primitive-manifolds and . For simplicity, we assume that all the groups have the canonical lattice of translations . Thus, by Theorem 2.1, all manifolds belonging to each family are mutually isospectral on forms. We now indicate the order of the cardinality of each family. We have that and (see [MP], [Ti]). By using a small subfamily of HW-manifolds it is proved in [MR] that . Furthermore, based on an example given in [LS], Rossetti constructed a very big family of GHW-manifolds with (see [MPR]).

3. Spin structures

Spin structures play a role in geometry and physics. One relevant fact is that they allow to define Dirac operators. On an arbitrary Riemannian manifold , the Laplacian on functions and the Laplacian on -forms are always defined. On the other hand, for the Dirac operator to be defined, needs to have an extra geometric structure. More precisely, to each spin structure on one can construct the so called spinor bundle and a Dirac operator acting on sections of it. However, one needs to have some care here, since not every Riemannian manifold admits a spin structure (see [LS]).

Let be an oriented Riemannian -manifold and let be the -principal bundle of oriented frames. Inside the group of units of the Clifford algebra of lies the compact connected Lie group

where . This group satisfies the exact sequence

where . Thus, is a double covering and since is simply connencted, for , it is the universal covering of .

A spin structure on an orientable manifold is an equivariant 2-fold covering such that , where is a -principal bundle. A manifold endowed with a spin structure is called a spin manifold.

Fortunately, for compact flat manifolds we can get rid of this complicated geometrical-topological definition by using the following result. The spin structures of are in a 1-1 correspondence (see [LM] or [Fr2]) with the group homomorphisms commuting the diagram

(3.1)

This gives a purely algebraic alternative definition, simpler than the original one, which is in fact a criterion to decide the existence of spin structures. It can be used not only to construct such structures, but also to count them.

Spin structures on-manifolds. In this subsection we refer to [MP]. Let be a -manifold and a spin structure as in (3.1). Since , then for any . Let be a -basis of and put . For , with , we have . For any we fix a distinguish (though arbitrary) element . Then,

(3.2)

where depends on and on the choice of .

The map is determined by its action on the generators of , and so we can identify it with the -tuple . Since is a group homomorphism it must satisfy, for every , the following conditions

Note that, since , these are conditions for over , i.e. for the character . We define the set

The following theorem says that the above necessary conditions 's for the existence of spin structures on -manifolds, are also sufficient .

defines a bijective correspondence between the spin structures ofand the set. Hence, the number of spin structures ofis eitherorforsome.

Applications. By applying Theorem 3.1 we can: (1) study the existence of spin structures in particular families of -manifolds, (2) give a simple method to obtain spin manifolds and (3) determine the audibility of the spin structures, that is, whether spin structures can be heard or not.

Spin structures in families.(i) Every orientable -manifold is spin and orientable -manifolds of diagonal type have spin structures (see [MP]), the same as for any -torus (see [Fr]). (ii) Orientable primitive -manifolds are spin. (iii) The 3-dimensional HW-manifold is spin. HW-manifolds of dimension , , are not spin. See [Po] for the case . The general case, i.e. odd, was proved independently by J. P. Rossetti ([Ro], by using a criterion in [MP]) and by S. Console ([Co], by computing the second Stiefel-Whitney classes ).

How to get spin manifolds easily?. By using the doubling procedure in [JR] or [BDM]. Let be an -dimensional Bieberbach group with translation lattice and holonomy group . The double of is the Bieberbach group defined by , with . It follows that d has translation lattice and holonomy group . The associated manifold has dimension and is Kähler (see [DM]). Now, doubling an orientable manifold of diagonal type gives a spin manifold (see [MP2]). If the manifold is not orientable, then one has to double twice.

Spin structures are not audible. Take and consider the manifolds and where and are Bieberbach groups of diagonal type given, in diagonal notation (see [MR2], [MP]), in the following table. For example, the in the first column means that . Also, and , .

It is easy to see that are orientable -manifolds of dimension 6 and that they are -isospectral for every , . Note that they are not primitive since . Using Theorem 3.1 we can check that has no spin structures while has spin structures of the form , with . Thus, we cannot hear the spin structures of Riemannian manifolds!

4. Dirac spectrum

We begin with the ingredients necessary to define twisted Dirac operators on Riemannian manifolds. Let be an orientable compact flat manifold endowed with a spin structure as in (3.1), denoted by from now on. Let be the spinrepresentation, that is the restriction to of any irreducible complex representation of the complexified Clifford algebra . It is wellknown that and that is irreducible if is odd while, if is even, splits into two inequivalent irreducible representations of the same dimension, called the half-spinrepresentations. Let be a unitary representation such that . As usual, we take and .

Now, the morphism allows to construct the spinor bundle twisted by

with action given by . One can identify the space of smooth sections of this bundle, , with the set .

With the above identification, the Dirac operator twisted by on compact flat manifolds is given by

where acts by in and is an orthonormal basis of . If we have the classical Dirac operator . is a first order elliptic differential operator, symmetric and essentially self-adjoint. It does not depend on the choice of the orthonormal basis of . Also, it is a formal square root of the Laplacian, that is , called the twisted spinor Laplacian. If , is called a harmonicspinor.

has a discrete spectrum consisting of real eigenvalues , , of finite multiplicity . Explicit expressions for for an arbitrary pair were obtained in [MP2]. We now recall this result.

Let where . Put , with the dual lattice of , and

Now, for each , let denotes the set of elements fixed by in , that is . Furthermore, for , let be a fixed (though arbitrary) element in the maximal torus of , conjugate in to . Finally, define a sign , depending on and on the conjugacy class of in , in the following way. If and , let such that . Hence, . Take if is conjugate to in and otherwise. As a consequence, and for every (see Definition 2.3, Remark 2.4 and Lemma 6.2 in [MP2] for details).

For odd, the multiplicity of the eigenvalue , for , is given by

(4.1)

while for even, it is given by the first term in (4.1), where the sum is taken over all , with replaced by . For , with even or odd, we have that , if , and , otherwise.

Dirac spectrum of-manifolds. In the particular case when , the formula (4.1) becomes more tractable and allows one to give shorter expressions for the multiplicities. Also, one can characterize all the spin -manifolds having asymmetric twisted Dirac spectrum. To wit

Theorem 4.1.Letbe an-dimensional spin-manifold.

(i) If, then the spectrumis symmetric and the non-zeroeigenvalueofhas multiplicity

(4.2)

(ii) If, thenis asymmetric if and only if:and thereexistswithandsuch that. In this case, theasymmetric spectrum is the set

We now deal with the isospectral problem for twisted Dirac operators . We claim the existence of twisted Dirac isospectral manifolds having different spectral, geometrical or topological properties. We will compare -isospectrality with other notions of isospectrality such as isospectrality with respect to the spinor Laplacian and the -Laplacian . We will also look at the spectrum of closed geodesic with and without multiplicities, that is the so called -spectrum and -spectrum, respectively.

Theorem 5.1.There are familiesof Riemannian-manifolds, pairwise nonhomeomorphic, which are mutually-isospectral for each, but they are neitherisospectral on functions nor-isospectral. Furthermore,can be chosen satisfyingany of the following extra properties:

(i) Everyhas (or has no) harmonic spinors.

(ii) All's inhave the same-Betti numbers forand they are-isospectral to each other for anyodd.

Theorem 5.3.There are big families of-isospectral manifolds. More precisely,there exists a familyof pairwise non-homeomorphic Riemannian-manifolds thatare all mutually-isospectral, for many different choices of spin structures, with thecardinality ofdepending exponentially onor, even better, on.

6. Eta series and eta invariants

Let be a self-adjoint elliptic differential operator of order on a compact -manifold . To study the spectral asymmetry of , Atiyah, Patodi and Singer introduced in [APS] the so called eta series defined by

This series converges for and defines a holomorphic function which has a meromorphic continuation to with simple poles (possibly) at , . It is a non trivial fact that this function is really finite at (See [APS2] for odd, [Gi] for even, and [Wod] using different methods). The number is a spectral invariant, called the eta invariant, which does not depend on the metric, although does. It gives a measure of the spectral asymmetry of and it is important because it appears in the "correction term" of some Index Theorems for manifolds with boundary. For example, if , the classical Dirac operator, and is a compact spin manifold with , under certain global boundary conditions, the index of is given by

where is the Hirzebruch -polynomial in the Pontrjagin forms , is the eta-invariant associated to , and (see [APS2]). Note the beauty of the above expression relating topological, geometrical and spectral data!

Eta invariants are not audible. There are 7-dimensional -manifolds which are -isospectral for such that but (see [Po]). The trick is to pick one manifold having symmetric spectrum while the other not. It turns out that 7 is the minimum dimension in which this can be done. The moral is that we cannot hear the -invariant of compact Riemannian manifolds!

Acknowledgements. I wish to thank the organizers of the "II Encuentro de Geometría" for having invited me to participate in the event, where so many good mathematicians were present. It was also a pleasure for me to write this extended version of the talk given there.